Answer
$\frac{a^{8}}{9b^{10}}$
Work Step by Step
Based on the power of a product rule, we know that $(ab)^{n}=a^{n}b^{n}$ (where $n$ is a positive integer and $a$ and $b$ are real numbers).
Therefore, $(\frac{3a^{4}}{9b^{5}})^{2}=\frac{3^{2}(a^{4})^{2}}{9^{2}(b^{5})^{2}}=\frac{9(a^{4})^{2}}{81(b^{5})^{2}}=\frac{(a^{4})^{2}}{9(b^{5})^{2}}$.
Based on the power rule for exponents, we know that $(a^{m})^{n}=a^{mn}$ (where $m$ and $n$ are positive integers and $a$ is a real number).
Therefore, $\frac{(a^{4})^{2}}{9(b^{5})^{2}}=\frac{a^{4\times2}}{9\times b^{5\times2}}=\frac{a^{8}}{9b^{10}}$.